Training Course on Basic Statistics for Research August 24-28, 2009 STATISTICAL RESEARCH AND TRAINING CENTER J and S Building, 104 Kalayaan Avenue, Diliman, Quezon City Measures of Location Prepared by: Josefina V. Almeda Professor and College Secretary School of Statistics University of the Philippines, Diliman August 2009

2 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 Learning Objectives After the session, participants should be able to:  To list and define the most common measures of location  To demonstrate and apply the use of measures of location;  Interpret results obtained from each measure.

3 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 Measures of Location  Percentiles  Quartiles  Deciles

4 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 * Percentiles divide the ordered observations into 100 equal parts. * There are 99 percentiles, denoted by P 1, P 2, P 3, …, P 99 with around 1% of the observations in each group. We read and interpret the individual percentiles as follows: P 1, read as first percentile, is the value below which 1% of the ordered values fall. P 2, read as second percentile, is the value below which 2% of the ordered values fall. : P 99, read as ninety-ninth percentile, is the value below which 99% of the ordered values fall. Percentiles

5 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 Thus, P k is a value such that at least k% of the ordered data are smaller than it and at least (100-k)% are larger than it, where k = 1, 2, 3, …, 99. For example, the 80th percentile of a distribution is a value such that at least 80 percent of the ordered observations are less than its value and at least 20 percent of the ordered observations are larger than its value.

6 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 How to Compute for Percentiles * arrange the data in ascending order before getting the percentile * assume that all observed values exist and that there is no missing data * let be the ordered observations arranged from lowest to highest * denote the percentile we are interested with k Example: if we want to compute the 75th percentile, then k = 75. If we want the 90th percentile, then k = 90.

7 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 Two Cases in Getting the Percentile: 1.Ifis an integer,P k = 2. If is not an integer, P k is the ith data item in the ordered observations where i is the closest integer greater than P k corresponds to the percentile that we want to find, where k = 1 to 99 and n is the number of observations. Empirical Distribution Number with Averaging

8 Statistical Research and Training Center Training Course on Basic Statistics for Research August , The annual per capita poverty threshold in pesos of the different regions of the Philippines are as follows: 15,693, 13,066, 12,685, 11,128 13,760, 13,657, 11,995, 11,372, 11,313, 9,656, 9,518, 9,116, 10,503, 10,264, 10,466, 10,896, 12,192. Find the 75th percentile. Examples of Getting the Percentile Using the Empirical Distribution Number with Averaging

9 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 Solution: Arrange the 17 annual per capita poverty threshold in pesos of the 17 regions of the Philippines from lowest to highest. Array: 9116, 9518, 9656, 10264, 10466, 10503, 10,896, 11128, 11313, 11,372, 11995, 12192, 12,685, 13066, 13657, 13760, 15,693 Compute for nk/100 where n = 17 and k = 75. nk/100 = 17(75)/100 = (not an integer) Since nk/100 is not an integer, we use the second formula in the empirical number distribution with averaging. The 75th percentile is 12,685. This implies that 75% of the 17 annual per capita poverty threshold falls below P12,685.

10 Statistical Research and Training Center Training Course on Basic Statistics for Research August , The following are the number of telephone lines of 16 regions for the year 2004: , 94079, , 42860, , , , , , , 35945, , , 82616, , Find the 50 th percentile. Solution: Arrange the observations from lowest to highest. Array: 33315, 35945, 42860, 82616, 94079, , , , , , , , , , ,

11 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 Compute for nk/100 where n = 16 and k = 50. nk/100 = 16(50)/100 = 8 (integer) Since nk/100 is an integer, we use the first formula of the empirical number distribution with averaging. Thus, 50% of the 16 regions have number of telephone lines lower than 149,582. P 50 =

12 Statistical Research and Training Center Training Course on Basic Statistics for Research August , value that divide an ordered data sets into 4 equal parts Split Ordered Data into 4 Quarters the i th quartile, Q i is a value below which 25x i % of the data lie 25% Q1Q1 Q2Q2 Q3Q3 = Quartiles

13 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 The Quartiles * The upper quartile denoted by Q 3 have the highest observed values of the data set. It divides the bottom 75% of the ordered observations from the top 25%. * The middle quartile denoted by Q 2 contains the next highest observed values of the data set. It divides the bottom 50% of the ordered observations from the top 50%. * The lower quartile denoted by Q 1 have the lowest observed values of the data set. It divides the bottom 25% of the ordered observations from the top 75%. Thus, Q k where k = 1,2,3 is a value such the k% of the ordered data are smaller in value than this.

14 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 * first quartile or lower quartile is the 25th percentile; * second quartile or the median is the 50th percentile; and * third quartile or the upper quartile is the 75th percentile. * Quartiles are special cases of percentiles. Thus, the formulas we have for the percentiles are applicable for the quartiles. Relationship of Quartiles and Percentiles

15 Statistical Research and Training Center Training Course on Basic Statistics for Research August , % Deciles  9 values that divide an ordered data set into 10 equal parts  The i th decile, D i is a value below which 10 x i % of the data lie D1D1 D2D2 D3D3 D4D4 D5D5 D6D6 D7D7 D8D8 D9D9

16 Statistical Research and Training Center Training Course on Basic Statistics for Research August , 2009 We read and interpret the deciles as follows: D 1, read as first decile, is the value below which 10% of the ordered values fall. D 2, read as second decile, is the value below which 20% of the ordered values fall. : D 9, read as ninth decile, is the value below which 90% of the ordered values fall. The Deciles

Training Course on Basic Statistics for Research August 24-28, 2009 STATISTICAL RESEARCH AND TRAINING CENTER J and S Building, 104 Kalayaan Avenue, Diliman, Quezon City Thank you.